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%I #13 Sep 08 2022 08:45:57
%S 1,4,7,8,3,1,8,3,4,3,4,7,8,5,1,5,9,5,6,4,2,2,1,0,4,4,3,6,3,8,5,0,2,2,
%T 2,1,5,2,5,3,2,1,2,1,1,5,0,4,9,9,0,6,4,1,6,7,0,8,4,0,3,9,1,0,2,6,4,9,
%U 9,8,0,5,4,3,7,0,5,7,3,3,2,3,3,6,7,5,1,8,8,2,0,7,4,0,8,2,1,3,6,6,9,7,8,1,0,9,6,7
%N Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).
%C The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+2x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes 1 and sqrt(2) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2,11,32,1,4,10,2,1,...] at A190261. For details, see A188635.
%H G. C. Greubel, <a href="/A190260/b190260.txt">Table of n, a(n) for n = 1..10000</a>
%e 1.478318343478515956422104436385022215253...
%t r=2^(1/2);
%t FromContinuedFraction[{1, r, {1, r}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A190261 *)
%t RealDigits[N[%%, 120]] (* A190260 *)
%t N[%%%, 40]
%o (PARI) (1+sqrt(1+2*sqrt(2)))/2 \\ _G. C. Greubel_, Dec 26 2017
%o (Magma) [(1+Sqrt(1+2*Sqrt(2)))/2]; // _G. C. Greubel_, Dec 26 2017
%Y Cf. A188635, A190262, A190258.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, May 06 2011
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